A (classical) solution of the wave equation $$ u_{tt}-c^2u_{xx}=0,\qquad (x,t)\in\mathbb{R}\times\mathbb{R}^*_+, $$ is required to be of class $C^2$. Why?
I mean, why would one impose that all second partial derivatives, even $u_{xt}$ , which does not appear in the PDE, must be continuous?!
You are right, it is some overkill. I guess you could require something like $$ u\big(x, \cdot\big) \in C^2\big(\mathbb{R}\big) \land u\big(\cdot, t \big) \in C^2\big(\mathbb{R}_+\big),$$ which is a minimal less restricted case although you probably just exclude some very specially tailored counterexamples.
